In the context of the DI theory, the cornerstone system for the motion of two viscous and incompressible fluids with matched (constant) densities is the model H, which was rigorously derived in [1]}. The model H consists of the following Navier–Stokes–Cahn–Hilliard system
\(&\rho \partial _{t}{v}+ \rho \textnormal {div}\big ( {v}\otimes {v}\big )- \textnormal {div}\big ( 2\nu (\phi ) \, \big )+ \nabla p= - \varepsilon \, \textnormal {div}\big (\nabla \phi \otimes \nabla \phi \big )&&\quad \text{in $Q$},\\[1ex]&\textnormal {div}\, {v}= 0&&\quad \text{in $Q$},\\[1ex]&\partial _{t}\phi + \textnormal {div}(\phi {v})= \textnormal {div}\big ( m_\Omega (\phi ) \nabla \mu \big )&&\quad \text{in $Q$},\\[1ex]&\mu = -\varepsilon \Delta \phi + \varepsilon ^{-1} F^{\prime }(\phi )&&\quad \text{in $Q$}.\) 
